A085812 Sum(sum(binomial(i,j),i=n..2*n),j=0..n).

1, 5, 22, 91, 366, 1454, 5748, 22691, 89590, 354010, 1400268, 5544334, 21973420, 87158972, 345977832, 1374249251, 5461704870, 21717305762, 86391846492, 343800647066, 1368639516420, 5450093895812, 21708897213912, 86492537630606

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 25 2003

Keywords

• nonn
• easy

Comments

As the definition indicates, each term is the sum of numbers from Pascal's Triangle in an (n+1) X (n+1) square arrangement.

Example for a(2):

1

1 1

|1 2 1 |

|1 3 3 |1

|1 4 6 |4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

...

and

(1+2+1) + (1+3+3) + (1+4+6) = 22 = a(2). Similarly, a(1) = (1+1)+(1+2) = 5. - John Molokach, Sep 17 2013

Examples

			a(1)=binomial(1,0)+binomial(2,0)+binomial(1,1)+binomial(2,1)=1+1+1+2=5
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Programs

Maple

a := n->add(add(binomial(i,j),i=n..2*n),j=0..n); seq(a(n),n=0..25);

Mathematica

Table[4^n-2^n+Binomial[2*n+2,n+1]/2,{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

PARI

a(n)=4^n-2^n+binomial(2*n+2,n+1)/2; \\ Joerg Arndt, May 10 2013

Formula

a(n) = 4^n - 2^n + C(2*n+2, n+1)/2. - Vaclav Kotesovec, Oct 28 2012

(n+1)*a(n) +2*(-6*n-1)*a(n-1) +4*(13*n-9)*a(n-2) +8*(-12*n+19)*a(n-3) +32*(2*n-5)*a(n-4)=0. - R. J. Mathar, Oct 01 2013